Macroeconomic Foundation of Monetary Accounting by Diagrams of Categorical Universals

We present a category theoretical formulation of the Monetary Macroeconomic Accounting Theory (MoMaT) of Men\'endez and Winschel [2025]. We take macroeconomic (national) accounting systems to be composed from microeconomic double-entry systems with real and monetary units of accounts. Category theory is the compositional grammar and module system of mathematics which we use to lift micro accounting consistency to the macro level. The main function of money in MoMaT is for the repayment of loans and not for the exchange of goods, bridging the desynchronisation of input and output payments of producers. Accordingly, temporal accounting consistency is at the macroeconomic level. We show that the accounting for macroeconomies organised by a division of labor can be consistent and stable as a prerequisite for risk and GDP sharing of societies. We exemplify the theory by five sectoral agents of Labor and Resource owners, a Company as the productive sector, a Capitalist for profits, and a Bank as the financial sector providing loans to synchronise the micro and the macro levels of an economy. The dynamics is described by eight sectoral macroeconomic bookings in each period demonstrating stable convergence of the MoMaT in numerical simulations. The categorical program implements a consistent evolution of hierarchical loan repayment contracts by an endofunctor. The universal constructions of a limit verify all constraints as the sectoral investment and learning function at the macroeconomic level. The dual colimit computes the aggregated informations at the macro level as usual in the mathematics of transitions from local to global structures. We use visual diagrams to make complex economic relationships intuitive. This paper is meant to map economic to categorical concepts to enable interdisciplinary collaboration for digital twins of monetary accounting systems.